Dynamic Rupture Simulation Methods Luis A. Dalguer Consultant at 3Q-Lab GmbH , Switzerland Visiting professor at DPRI, Kyoto University, Japan Adjunct Professor at Aichi Institute of Technology, Japan
Content • Introduction: Fault complexity (Earthquakes occur on faults) • Idealization of faulting for rupture dynamic • Some types of friction laws • Problem statement for rupture modeling • Mathematical representation of earthquake for rupture dynamic • Geometrical consideration of faults for modeling • Numerical techniques for rupture dynamic • Fault representation methods for dynamic rupture simulation • Numerical resolution to solve rupture dynamic • Assessment of Fault representation Methods
Introduction: fault complexity Faults in nature Earthquakes occur on Faults: But how do they look like? • Faults are not isolated (segmented and linked, irregular and rough at all scales)
Introduction: fault complexity Schematic map views of fault structures at different scales Primary slip surface (Ben-Zion and Sammis, 2003)
Introduction: fault complexity How faults may looks at depth and shallow? (Ben-Zion et al, 2007)
Idealization of faulting for rupture dynamic All is about cracks: Ø When active during earthquakes, dominantly operate as dynamically running shear cracks Ø Then it is in principle a Fracture Mechanics problem Ø Fracture Mechanics: Quantitative description of the mechanical state of a deformable body containing a crack or cracks. Ø Then Dynamic Rupture Models have their foundation in Fracture mechanics concepts. Ø Dynamic models usually idealize the earthquake rupture as a dynamically running shear crack on a frictional interface embedded in a linearly elastic and/or non linear continuum. Ø Incorporation of small scale complexities in numerical simulations requires high resolution models
Idealization of faulting for rupture dynamic Cohesive z zone ( (Fracture m mechanics) a and f friction m model • Models -Constant (Barenblatt, 1959) -Linearly dependent on distance to crack tip (Palmer and Rice, 1973; Ida, 1973) -Linearly dependent on slip (Ida, 1973 Andrews; 1976) τ s For the scale of earthquake modeling, G c is a mesoscopic parameter, contains all the dissipative processes in the volume around the Crack tip ξ crack tip: off-fault yielding, damage, micro- τ d Λ cracking etc. Cohesive zone -They are mapped on the fault plane.
Idealization of faulting for rupture dynamic Stress and friction on the fault (crack) The earthquake rupture can be described as a two-step process: (1) formation of crack and (2) propagation or growth of the crack. The crack tip serves as a stress concentrator due to driving force; if the stress at the crack tip exceeds (The cohesive zone: break down process that needs to some critical value, then the crack grows unstably accompanied by a sudden be accurately solved) slip and stress drops. Stress concentration τ y = Yielding stress Slip Stress on fault Fault rupture Friction sliding Crack tip (Rupture front)
Friction laws: Slip weakening τ c τ c Thermal pressurization? τ s Fracture Energy τ s Fracture Energy g >> τ 0 s τ 0 (Ida, 1972; Andrews, 1976) G c G c τ d τ d d 0 Slip (s) d 0 Slip (s) gt - t + g t - t ì ( d s ) s d ( ) ( d s ) s + < g >> » + s 0 d 0 s d for s s 0 d ï + g + g 0 t = í d ( s ) d ( s ) d d 0 0 0 0 c ï t ³ s d î d 0 Input requirement: g = g >> » Define No-linearity s then Linear weakening t = t = t = Initial shear stress, Static friction, Dynamic friction 0 s d = d Critical slip-weakening 0
Friction laws: Rate and state Aging law (Dieterich, 1986; Ruina, 1983) (its basis in laboratory experiments) ( ) ( ) t q = s é + + q ù ( , ) V f a ln V V b ln V L ë û c n 0 0 0 q = - q 1 V L q y = and State variables = = = V Slip rate ( V steady state reference, V weakening) 0 w = = = = f Friction coeficient ( f steady state, f at steady state V , f weakening) ss 0 0 w = a b , Friction parameters
Friction laws: Rate and state ( ) t q = s é + y ù ( , ) V a ln V V Slip law ë û c n 0 - V [ ] y = - - f f ( ) V ss L ( ) = - - f ( ) V f ( b a )ln V V ss 0 0 Strong velocity weakening (Flash heating): same as slip law, but Motivated by high-speed rock sliding experiments (e.g., Tsutsumi and Shimamoto, 1997; Di Toro et al., 2004; Han et al., 2010; Goldsby and Tullis, 2011) ( ) ì = - - < f f ( b a )ln V V if V V ï ss aging ( ) 0 0 w = í f ( ) V ss é ù + - > f f f V V if V V ï ë û î w ss aging ( ) w w w q y = and State variables = = = V Slip rate ( V steady state reference, V weakening) 0 w = = = = f Friction coeficient ( f steady state, f at steady state V , f weakening) ss 0 0 w = a b , Friction parameters (La Pusta et al., 2000; Noda et al., 2009; Rojas et al., 2009; Dunham et al., 2011; Shi and Day, 2013)
Problem statement for rupture modeling Volume domain of interest (a piece of the earth) Fault (a discontinuity in the earth)
Problem statement for rupture modeling σ 1 Tectonic loading σ 3 Stress concentration τ Fault rupture (Dynamically propagates as a running shear crack) σ 1 σ 3
̇ Mathematical representation Elastodynamic coupled to frictional sliding (Highly non-linear problem) ! ̇ # $ = & ' ( $' ( $' = ) $' *+ & * # + τ ≤τ c , ≤ . / = 0(( 2 , 4, ̇ 4, 5 6 , 5 7 … Friction constitutive equation
Mathematical representation Fault-surface boundary conditions For shear (nonlinear) For opening (nonlinear) s ³ 0 n ! − ! # ≤ % ³ U 0 n s = U 0 n n
Geometrical representation of faults for modeling Simplification of fault geometry for earthquake dynamic (depending of numerical method: FDM, FEM, SEM,DG,BIEM
Geometrical representation of faults for modeling Simplification of fault geometry for earthquake dynamic (depending of numerical method: FDM, FEM, SEM,DG,BIEM
Geometrical representation of faults for modeling Simplification of fault geometry for earthquake dynamic (depending of numerical method: FDM, FEM, SEM,DG,BIEM
Numerical techniques for rupture dynamic scecdata.usc.edu/cvws
Numerical techniques for rupture dynamic http://scecdata.usc.edu/cvws/code_descriptions.html Spontaneous Rupture Code Descriptions (There are also other codes outside of this project)
Fault representation methods for numerical simulation • Traction at Split-node method Fault Discontinuity explicitly incorporated (Andrews, 1973; DFM model: Day, 1977, 1982; SGSN model, Dalguer and Day, 2007) • “Inelastic-zone” methods: Fault Discontinuity not explicitly incorporated - Thick-fault method (TF) (Madariaga et al., 1998) - Stress-glut (SG) method (Andrews 1976, 1999)
Fault representation methods for numerical simulation Traction at Split-Node method For Staggered Grid For partially Staggered Grid Staggered-Grid Split-Node Method (SGSN) (e.g, model DFM (Dalguer and Day 2007, JGR) Day, 1982; Day et al, 2005)
Fault representation methods for numerical simulation Central Differencing in time (representation of equation of motion on fault (Slip velocity) ! " Compute “trial” traction (enforces continuity of tangential velocity # and continuity of normal displacement. Then actual nodal traction (tangential components n =x,y) " #
Fault representation methods for numerical simulation “Inelastic-zone” Fault models (in Staggered Grid FDM) (Dalguer and Day, 2006, BSSA) Thick-fault method (TF) Stress-glut method (SG) (Madariaga et al., 1998) (Andrews 1976, 1999)
Fault representation methods for numerical simulation “Inelastic-zone” Fault models s Nodal Stress by Central Differencing in time gives (example ) xz addition of an inelastic component to the total strain rate ! " Compute “trial” traction setting # Then set the fault plane traction to
Fault representation methods for numerical simulation Stress-glut method (SG) (Andrews 1976, 1999) Frictional bound enforced on one plane of traction nodes Calculate inelastic component Calculate the total slip rate by D integrating over the spatial step x Thick-fault method (TF) (Madariaga et al, 1998) -Frictional bound enforced on 2 planes of traction nodes -Slip-velocity given by velocity difference across 2 unit-cell wide zone
Numerical resolution to solve rupture dynamic The cohesive zone for a slip-weakening crack T s =Static yielding stress; T d =Dynamic yielding stress T 0 =Initial shear stress; d 0 =Critical slip distance ∆" = T 0 - T d = Stress drop Ø At the scale of natural earthquakes, the cohesive zone examines the crack tip phenomena at a level of observation, in which the fracture energy Gc is a mesoscopic parameter which contains all the dissipative processes in the volume around the crack tip, such as off-fault yielding, damage, micro-cracking, etc. Ø In the cohesive zone, shear stress and slip rate vary significantly and proper numerical resolution of those changes is crucial for capturing the maximum slip rates and the rupture propagation time and speeds. Therefore, the cohesive zone developed during rupture propagation need to be accurately solved to obtain reliable solution of the problem.
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